Question

Knife blades are often made of hardened carbon steel. The hardening process is a heat treatment in which the blade is first heated to a temperature of ${1346}^{\circ }\mathrm{F}$ and then cooled down rapidly by immersing it in a bath of water. To achieve the desired hardness, after heating to ${1346}^{\circ }\mathrm{F}$, a blade needs to be brought to a temperature below $5.00\cdot {10}^2{}^{\circ }\mathrm{F}$. If the blade has a mass of $0.500\mathrm{kg}$ and the water is in an open copper container of mass $2.000\mathrm{kg}$ and sufficiently large volume, what is the minimum quantity of water that needs to be in the container for this hardening process to be successful? Assume the blade is not in direct mechanical (and thus thermal) contact with the container, and neglect cooling through radiation into the air. Assume no water boils but reaches ${100}^{\circ }\mathrm{C}.$ The heat capacity of copper around room temperature is $c_{\text{copper }}=386\mathrm{J}/(\mathrm{kg}\mathrm{K}).$ Use the data in the table below for the heat capacity of carbon steel

Short answer

Answer: The minimum amount of water required is approximately 0.1396 kg.

Step-by-step solution

1. Convert temperature to Celsius

We need to convert the initial and final temperatures of the knife into Celsius since the heat capacity data is usually expressed in Celsius. To convert from Fahrenheit to Celsius, we use the formula: $$T_C=(T_F-32)\frac{5}{9}$$. Applying the formula to the initial temperature of the blade (${1346}^{\circ }\mathrm{F}$): $$T_{C_i}=(1346-32)\frac{5}{9}=\boxed{\text{723.3^circmathrm{C}}}$$. And to the final temperature of the blade ($\text{Double exponent: use braces to clarify}$): $$T_{C_f}=(500-32)\frac{5}{9}=\boxed{\text{260.0^circmathrm{C}}}$$.

2. Calculate the heat lost by the knife

Using the heat capacity of carbon steel (c = 470 J/kg·C, given in the text) and the temperature change calculated above, we can calculate the heat ($Q_{knife}$) lost by the knife: $$Q_{knife}=m_{knife}{c}_{steel}\mathrm{\Delta }{T}_{knife}$$, where $m_{knife}=0.500\mathrm{kg}$, $c_{steel}=470\mathrm{J}/\mathrm{kg}·\mathrm{C}$, and $\mathrm{\Delta }{T}_{knife}=T_{C_i}-T_{C_f}=723.3-260.0={463.3}^{\circ }\mathrm{C}$. $$Q_{knife}=(0.500)(470)(463.3)=\boxed{\text{108485.5 ,mathrm{J}}}$$

3. Calculate the heat gained by the container

Since the knife does not have direct contact with the container, the heat will not transfer directly. The heat capacity of copper around room temperature is given as $c_{copper}=386\mathrm{J}/(\mathrm{kg}\cdot \mathrm{K})$. We assume that in the process, the copper container heats up to ${100}^{\circ }\mathrm{C}$ (the boiling point of water). We first need to calculate the heat gained by the container to raise its temperature to ${100}^{\circ }\mathrm{C}$. Assume the container to have a room temperature of 20°C. $$Q_{container}=(2.000)(386)(100-20)=\boxed{\text{61840 ,mathrm{J}}}$$

4. Calculate the heat gained by the water

Subtract the heat gained by the copper container found in step 3 from the heat lost by the knife to find out how much heat the water has to gain: $$Q_{water}=Q_{knife}-Q_{container}=108485.5-61840=\boxed{\text{46645.5 ,mathrm{J}}}$$

5. Calculate the mass of the water

Now, we can use the heat equation in reverse to determine the water's mass. Given the heat capacity of water $c_{water}=4186\mathrm{J}/(\mathrm{kg}\cdot \mathrm{C})$, the change of temperature of water is $\mathrm{\Delta }{T}_{water}={100}^{\circ }\mathrm{C}$ - 20°C = 80°C. $$Q_{water}=m_{water}{c}_{water}\mathrm{\Delta }{T}_{water}$$, Rearrange the equation to find $m_{water}$: $$m_{water}=\frac{Q_{water}}{c_{water}\mathrm{\Delta }{T}_{water}}$$ Using the values $Q_{water}=46645.5\mathrm{J}$, $c_{water}=4186\mathrm{J}/\mathrm{kg}\cdot \mathrm{C}$, and $\mathrm{\Delta }{T}_{water}={80}^{\circ }\mathrm{C}$: $$m_{water}=\frac{46645.5}{4186\cdot 80}=\boxed{\text{0.1396 ,mathrm{kg}}}$$ The minimum quantity of water required to successfully complete the hardening process is approximately $0.1396\mathrm{kg}$.

Key concepts

Heat Capacity

Heat capacity is a fundamental property that quantifies the amount of heat required to change a substance's temperature by a given amount. It is often denoted by the symbol 'c' and is measured in units of joules per kilogram per degree Celsius (J/kg·°C) or joules per gram per degree Celsius (J/g·°C). It's essential to understand that heat capacity is an extensive property, meaning it depends on the amount of the material.

Different materials have different heat capacities. For example, metals typically have lower heat capacities than water, meaning they require less heat to increase their temperature. This property is crucial in various applications, such as the hardening process of a carbon steel blade described in the exercise. The heat capacity of carbon steel, given as 470 J/kg·°C, tells us how much energy is required to raise the temperature of a certain mass of the steel blade by one degree Celsius.

In practical terms, if you have a material with a high heat capacity, it will take more energy to change its temperature, making it an excellent choice for applications that require thermal stability. Conversely, materials with lower heat capacities can quickly heat up or cool down, which is ideal in situations where rapid temperature changes are desired.

Temperature Conversion

Temperature conversion is necessary when dealing with heat transfer calculations, as temperatures can be measured in different scales, namely Celsius (°C), Fahrenheit (°F), and Kelvin (K). The exercise provided demonstrates the conversion from Fahrenheit to Celsius, which is important because the heat capacity of materials is usually given in Celsius or Kelvin.

The formula to convert Fahrenheit to Celsius is given as: $$T_C=(T_F-32)\frac{5}{9}$$.
To understand this conversion, it's helpful to know the freezing point of water is 32°F (0°C), and the boiling point is 212°F (100°C). The difference in these two benchmarks is 180°F or 100°C, which explains the fraction 5/9 in the conversion formula. For students, mastering this conversion is important to ensure that calculations are accurate and consistent with the units of heat capacity.

In the exercise, knowing how to accurately convert these temperatures is essential to understand how much energy the steel blade will release or absorb during the hardening process.

Thermal Energy Calculation

Thermal energy calculation involves determining the amount of heat transfer involved in a process. The basic formula used in such calculations is: $$Q=mc\mathrm{\Delta }T$$,where 'Q' represents the heat energy in joules, 'm' is the mass of the substance, 'c' is the heat capacity, and '\Delta T' is the change in temperature.

In the context of the exercise, this formula helps to calculate the heat (Q) lost by the steel blade and the heat gained by the copper container and the water. Once we know the heat lost by the blade, it becomes crucial to ensure that the water has enough capacity to absorb that heat to achieve the required hardening. The heat gained by the water is the heat lost by the blade minus the heat absorbed by the copper container.

The final step is to use the rearranged heat equation to calculate the mass of water needed, which considers the heat to be absorbed, the heat capacity of water, and the temperature change. Understanding thermal energy calculation is vital for students to apply the concept of heat transfer to real-world scenarios efficiently.